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We show that, given integers $n_1,n_2, \ldots ,n_k$ with $2 < n_1 < n_2 <\cdots < n_k$, there exists a local (Noetherian) unique factorization domainthat has maximal chains of prime ideals of lengths $n_1, n_2, \ldots ,n_k$which are disjoint except at their minimal and maximal elements. In addition,we demonstrate that unique factorization domains can have other unusual primeideal structures.

Maximal Chains of Prime Ideals of Different Lengths in Unique Factorization Domains